Monotonic series for fractions near π and their convergents

Author:

Lucas Stephen K.,Nimbran Amrik Singh

Abstract

We describe various methods to derive monotonic infinite series for fractions near π and obtain a variety of series for the special case of its convergents. These series immediately show that π is clearly different from these fractions, replicating with series the results in Dalzell [1, 2] and Lucas that used integrals with non-negative integrands to represent the gaps between π and fractions.

Publisher

Cambridge University Press (CUP)

Subject

General Mathematics

Reference15 articles.

1. Continued Fractions

2. 3. Lucas, S. K. , Approximations to π derived from integrals with nonnegative integrands, Am. Math. Monthly 116(2) (2009) pp. 166–172.

3. 8. Beukers, F. , A rational approach to π, Nieuw Archief voor Wiskunde, (December 2000) available at http://www.nieuwarchief.nl/serie5/pdf/naw5-2000-01-4-372.pdf

4. Diophantine Approximations and Diophantine Equations

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